According to Markov chain convergence theorem [1]}, an irreducible and aperiodic Markov chain on finite state space converges to a unique stationary distribution \(\pi \) .
The mixing time or mixing rate measures the rate at which the chain converges.
For two distributions \(\mu ,\nu \)  over the same sample space \(\Omega \) , the total variation distance between \(\mu \)  and \(\nu \)  is defined as
\({d_{\mathrm {TV}}}(\mu ,\nu ) \triangleq \frac{1}{2}\sum _{x\in \Omega }|\mu (x)-\nu (x)|.\) 
